Geometric Sequences

A geometric sequence is a sequence of numbers in which the ratio between consecutive terms is constant.

We can write a formula for the n^th term of a geometric sequence in the form

a_n = crⁿ,

where r is the common ratio between successive terms.

Example 1:

{2, 6, 18, 54, 162, 486, 1458, ...}

is a geometric sequence where each term is 3 times the previous term.

A formula for the n^th term of the sequence is

Example 2:

is a geometric series where each term is –1/2 times the previous term.

A formula for the n^th term of this sequence is

Example 3:

{1, 2, 6, 24, 120, 720, 5040, ...}

is not a geometric sequence. The first ratio is 2/1 = 2, but the second ratio is 6/2 = 3.

No formula of the form

a_n = crⁿ can be written for this sequence.